QUESTION 4 (20 Marks) Temperature in a plate as shown in the following Figure is governed by the Laplace equation within 0?x?2 and 0?y?2
$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$
$\frac{\partial T}{\partial y} = -10$
$\frac{\partial T}{\partial x} = 150$
The temperature at the left edge is fixed at 10°C, and that at the central and right nodes of the bottom edge is fixed at 40°C as shown in the Figure. The nodes 3 and 4 are subject to the temperature gradient of $\frac{\partial T}{\partial y} = -10$, and the nodes 2 and 4 are
subject to the temperature gradient $\frac{\partial T}{\partial x} = 150$. The central finite difference formula is requested to be applied for boundary nodes.
a) Derive finite difference equations for nodes 1 - 4 with ?x = ?y = 1.0. (12 marks) (tips: node 4 are subjected to two gradient conditions simultaneously)
b) Assuming the initial temperature of the nodes 1-4 are 20 °C, please calculate the temperature values of nodes 1-4 using Liebmann's method with a relaxation factor ? = 1.5 (First iteration only). (8 marks)
